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<p>Computer algebra packages like Mathematica, Matlab and Maple know Laplace Transforms of all the functions you are likely to encounter, so you have access to these online, and the packages have also an inversion routine to find a function <span class="process-math">\(f\)</span> from a given <span class="process-math">\(F.\)</span> There are books with long lists of transforms of known functions and compositions of functions; we give some in Section <code class="code-inline tex2jax_ignore">[cross-reference to target(s) "table" missing or not unique]</code>, which you should read through, eg some that are harder to calculate:</p>
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\begin{equation*}
{\mathcal L}[t^n] = \frac{n!}{s^{n+1}}, \ n=0,1,2,\ldots, \ \
{\mathcal L}[t^{1/2}] = \frac{1}{2}\left( \frac{\pi}{s^3} \right)^{1/2}, \ \
{\mathcal L}[t^{-1/2}] = \left( \frac{\pi}{s} \right)^{1/2}.
\end{equation*}
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